So choosing some arbitrary staring value located in region three I can "trace down" to the x-axis along a line parallel to either of the the blue lines (right or left) to the x-axis. So you have two ways of doing this (1) following 'down & left' will land in the u=1 region, but tracing 'down & right' (paralleling the "-ct" line) lands you outside of the u=1 region where u=0 there...using the eqn for u(x,t) (at 9:10) you'll avg out to u=1/2 for region 3, as noted in the video.
u = 0 in region four because as you follow down along the characteristic lines to the x-axis (via the left or right blue line) both options lead you outside of the region where u = 1 (x=-1 to x=+1). The "game" is to follow down along these characteristic lines from your region of interest (regions: 1-6) which parallel the blue lines drawn. Therefore these blue lines represent "boundary conditions" for the given Initial Value Conditions which are.
When I write it like that, it sounds like a lot! I imagine you could talk about shockwaves separately from Burger's, but I'm not sure, since I've only ever seen those topics thrown together. I guess what I'm interested in is more shockwaves/rarefaction waves and the entropy condition. Anyway, thanks for the response! Thanks for all the great videos!
I'd personally like to see you do an initial value problem with the inviscid Burger's equation, where the initial values are defined using an equation that has some jump discontinuities; like U(x,0)=-1 , x≤-1 and U(x,0)=1 , -1
"the lines that demarcate region 3 also go outside of the region where u = 1, yet the value of u in that region is certainly not 0."... True, but the "game" is not about what value do the given function for u have at the initial point x0... the game is what value does 'u' have when I intercept the x-axis by tracing down along a line parallel to the blue lines (by tracing to the right or to the left)... note tracing down right/left to the x-intercept signifies "x0+/- ct"
"Does region 4 not encompass exactly the same x interval as region 1?" In terms of the fact that region 4 is directly above the same points along the horizontal (x-axis) as region 1.... yes that's true. But in terms of where do I intercept the horizontal axis (a.k.a x-axis) if I follow along a line which parallels the blue characteristic lines originating in region 4, then any line you're following (either to the right or left) will intercept outside of the u=1 region (x = +/-1) if parallel.
I think that is because we are integrating from the left point to the right point, and region 4 covers the entire area. This is different from the f(x) case due to integration.
Hello everyone. Can someone explain to me why the slope of the characteristics lines is 1/c and -1/c? (We see them in minute 2:42) I would appreciate any help. Thanks in advance
Could you do the same thing if f and g are functions of (x1,x2)? Take the average of f along a circle in the x1/x2 plane with radius t. Then take the average of g inside that circle?
I really appreciate your posts on PDE on youtube. Thanks for these. But I've run into an issue with understanding the significance of characteristic equations. Are their sole purpose to define "lines" in the x vs t with constant "phase" for the wave? If so them I'm a bit lost as to the purpose of the "x+/- ct" lines..... In and of themselves I understand how you devised the "x+/- ct" lines but as to their purpose in regards to characteristic properties I'm not following. Can you enlighten me plz
These are fantastic! I'd love to see more, especially focusing on the Laplace and Burger equations!
So choosing some arbitrary staring value located in region three I can "trace down" to the x-axis along a line parallel to either of the the blue lines (right or left) to the x-axis. So you have two ways of doing this (1) following 'down & left' will land in the u=1 region, but tracing 'down & right' (paralleling the "-ct" line) lands you outside of the u=1 region where u=0 there...using the eqn for u(x,t) (at 9:10) you'll avg out to u=1/2 for region 3, as noted in the video.
u = 0 in region four because as you follow down along the characteristic lines to the x-axis (via the left or right blue line) both options lead you outside of the region where u = 1 (x=-1 to x=+1). The "game" is to follow down along these characteristic lines from your region of interest (regions: 1-6) which parallel the blue lines drawn. Therefore these blue lines represent "boundary conditions" for the given Initial Value Conditions which are.
Intuition into derivation, how it's supposed to be done. Props!
Thanks a lot this lesson really help clear things up. I hope you keep making them. Cause I will need these to pass my PDE course!
When I write it like that, it sounds like a lot! I imagine you could talk about shockwaves separately from Burger's, but I'm not sure, since I've only ever seen those topics thrown together. I guess what I'm interested in is more shockwaves/rarefaction waves and the entropy condition. Anyway, thanks for the response! Thanks for all the great videos!
Amazing explanation as always...Thank you sir !!!
I'd personally like to see you do an initial value problem with the inviscid Burger's equation, where the initial values are defined using an equation that has some jump discontinuities; like U(x,0)=-1 , x≤-1 and U(x,0)=1 , -1
I love the explanation! Would you also post a video on how to graph the time snap shots?
Thank you very much for this great video, it was really helpful in clarifying the wave equation.
Very helpful sir! Much thanks.
"the lines that demarcate region 3 also go outside of the region where u = 1, yet the value of u in that region is certainly not 0."... True, but the "game" is not about what value do the given function for u have at the initial point x0... the game is what value does 'u' have when I intercept the x-axis by tracing down along a line parallel to the blue lines (by tracing to the right or to the left)... note tracing down right/left to the x-intercept signifies "x0+/- ct"
"Does region 4 not encompass exactly the same x interval as region 1?" In terms of the fact that region 4 is directly above the same points along the horizontal (x-axis) as region 1.... yes that's true. But in terms of where do I intercept the horizontal axis (a.k.a x-axis) if I follow along a line which parallels the blue characteristic lines originating in region 4, then any line you're following (either to the right or left) will intercept outside of the u=1 region (x = +/-1) if parallel.
what is region 4 for when g(x) is box(x)? if you trace back the lines doesn't it go into the regions where g(x) is zero? how come it is 1/c
I think that is because we are integrating from the left point to the right point, and region 4 covers the entire area. This is different from the f(x) case due to integration.
Okay thank you, that makes a lot more sense.
Why is region 4 at 12:25 equivalent to u = 1? Aren't the bounds of integration from -1 to 1?
Hello everyone.
Can someone explain to me why the slope of the characteristics lines is 1/c and -1/c? (We see them in minute 2:42)
I would appreciate any help. Thanks in advance
Could you do the same thing if f and g are functions of (x1,x2)?
Take the average of f along a circle in the x1/x2 plane with radius t. Then take the average of g inside that circle?
I really appreciate your posts on PDE on youtube. Thanks for these. But I've run into an issue with understanding the significance of characteristic equations. Are their sole purpose to define "lines" in the x vs t with constant "phase" for the wave? If so them I'm a bit lost as to the purpose of the "x+/- ct" lines..... In and of themselves I understand how you devised the "x+/- ct" lines but as to their purpose in regards to characteristic properties I'm not following. Can you enlighten me plz
so how do you plot the lines
i love you bro
¿What software do you use? Great video thank you
i think its Mathematica (spelling may be wrong)
why you dont showing example with g(x) that not zero and f(x) not zero too. ?
Because that just gives u = 0 which is nothingness
what is this program?